Invariant polynomials on truncated multicurrent algebras

Abstract

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form g ⊗ F F[t1, . . . , t ℓ ]/I, where g is a finite-dimensional Lie algebra over a field F of characteristic zero, and I is a finite-codimensional ideal of F[t1, . . . , t ℓ ] generated by monomials. In particular, when g is semisimple and F is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in g ⊗ F F[t1, . . . , t ℓ ]/I. As an application of our main result, we show that the center of the universal enveloping algebra of g ⊗ F F[t1, . . . , t ℓ ]/I acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.

Casimir elements and Sugawara operators for Takiff algebras

“…This result was extended by Raïs and Tauvel [18] to all values of ℓ. More recently, Macedo and Savage [12] proved its multi-parameter generalization, while Panyushev and Yakimova [17] showed that this generalization remains valid for a wide class of Lie algebras g beyond simple Lie algebras.…”

Takiff Algebras With Polynomial Rings of Symmetric Invariants

“…(This goes back to Chevalley and Kostant.) Therefore, Theorem 0.1 yields another proof and a generalisation of [MS16,Theorem 5.4], see Corollary 2.6. A notable difference between our Theorem 0.1 and results of [AP17] is that we do not impose a constraint on i deg f i , which is a part of the definition of a "good generating system", and do not require the codim-2 property for q (see Section 1 for the definition).…”

“…, r. It then follows from (2·1) and the iteration process (2·2) that ξ = (ξ [i 1 ,...,ir] ) ∈q * reg ⇐⇒ ξ [m 1 ,...,mr] ∈ q * reg (see also Prop. 4.1(b) in [MS16]). Assume that q satisfies all the assumptions of Theorem 2.2 and set N = π −1 Q (π Q (0)) ⊂ q * .…”

“…. (m r + 1)·rk g, see [MS16]. The proofs heavily use the fact that g is semisimple, when many structure results are available.…”

“…The proofs heavily use the fact that g is semisimple, when many structure results are available. For instance, both [RT92] and [MS16] exploit Kostant's section for the set of the regular elements of g. On the other hand, if g is simple and q = g e is the centraliser of a nilpotent element e ∈ g such that g e has the "codim-2 property" and e admits a "good generating system" in k[g] G , then k[g e m * ] ge m is a polynomial ring for all m ∈ N, see [AP17,Theorem 3.1]. In all these cases, the free generators of the ring of symmetric invariants ofĝ or g e m are explicitly described via those of g or g e , respectively.…”